Program

There is much talk about outliers in data, and the identification of outliers is often important in practice. I will talk about the problem of defining the outlier identification problem. It is possible to define outliers data analytically (as implied for example by the definition of boxplots), but in order to assess formally whether and to what extent any method does a good job identifying outliers, a model-based definition is required. Outliers can be defined with respect to a model for non-outliers. A widespread approach is to investigate contamination mixtures where the majority of observations is assumed to be generated by the non-outlier model (often but not necessarily the Gaussian distribution), and outliers are generated by a “contaminating” minority component. Davies and Gather [1] argued that outliers should be defined as unlikely observations based on the reference distribution for non-outliers alone; not allowing any “contamination component” to generate “outliers” that are not actually “outlying” w.r.t. the reference distribution. Besides discussing these concepts, their implications, and the choices the researcher needs to make in order to apply them, I will discuss how outliers can be defined w.r.t. a reference distribution, following Davies and Gather, in such a way that “true outliers” are defined for general distributions not necessarily involving the reference distribution as a mixture component, which is arguably too restrictive for real data generating processes. If time allows I will comment on specific problems such as distributional skewness, outliers in cluster analysis, nonparametric outlier definition based on low density regions.

The tclust package for the R environment for statistical computing implements different robust non-hierarchical clustering algorithms where trimming plays a key role [2; 5]. The current 2.0 version of the package implements a number of new features, like improved initialization routines, parallel computing, maximization of constrained mixture loglikelihood, robust linear grouping and others, but the most exciting new feature is the automatic selection of the number of groups k, the constraint constant c and possibly the trimming factor alpha. The implementation is based on the new approaches for selecting k, c and alpha proposed by [1] and [4]. A new methodology for constrained parsimonious model-based clustering which includes fourteen parsimonious models that are often applied in model-based clustering when assuming normal components as limit cases is extended to cope with the presence of atypical observations [3]. Useful graphical tools for summarizing the clustering output and assisting with the choice of the number of groups and the other parameters are provided. A series of examples based on real and simulated data with different degrees of overlap will be given to demonstrate the performance of the new implementation of the different proposals.

Robust methods protect inference against contamination and can indicate subsets of the data with distinct structures. But applied conclusions depend on the robustness–efficiency tuning. The monitoring approach replaces a single robust fit with a sequence of fits indexed by subset size or breakdown point [1] and tracks residuals, coefficients and test statistics along the sequence. Following the order of the verbal presentation, we describe reproducible software, the monitoring with robust regression in a large banking study, finite-sample size and power of outlier tests, and monitoring-based choice of Box–Cox transformations [3] via fan plots. The message we want to convey is that monitoring gives applied robust analysis a reproducible reporting language. Instead of quoting only a coefficient table with t-statistics, an outlier list or a selected transformation, the analyst can report the interval of breakdown points or subset sizes over which the conclusion remains stable [2]. This is particularly useful when the data contain more than one population, because a change in the trajectories can reveal where a robust fit changes from protecting the main group to absorbing a secondary structure. In this sense, monitoring is both a robustness device and an exploratory diagnostic for model adequacy. Overall, monitoring turns robust statistics into a diagnostic workflow. It links outlier detection, tuning of robustness and transformation choice within a single graphical and computational framework. The approach is implemented in reproducible software through the FSDA toolbox for MATLAB, and the examples considered here emphasize that robustness should be reported dynamically: conclusions are most credible when they produce stable results that persist across a meaningful interval of subset sizes or breakdown-point values.

Outliers, or atypical observations, pose a significant challenge in statistical analysis. Even a few unusual data points can distort estimates and lead to misleading statistical conclusions. Robust methods have been developed to reduce the influence of such observations, ensuring reliable inference while maintaining efficiency when the data largely conform to assumed models. While often seen as problematic, outliers can also carry valuable information and be of intrinsic interest. In many applied settings, detecting atypical observations is not merely a preparatory step but a central analytical goal, as they may signal important events, novel findings, or remarkable changes. Developing effective methods to identify and display these anomalies is a major focus of current statistical research. Real-world datasets do not always originate from a single homogeneous population; instead, multiple overlapping groups often underlie the data. In such contexts, clustering provides a natural framework for uncovering latent structure. However, these methods are typically sensitive to outliers, which can severely distort cluster identification. This motivates the use of robust clustering approaches that can simultaneously accommodate heterogeneity and resist the influence of atypical observations. Robust clustering methods, particularly those based on trimming [1], offer a practical solution for simultaneously dealing with heterogeneity in the data and the possible presence of anomalies. Among these methods, TCLUST [2] stands out as an interesting approach with well-established theoretical and practical properties. However, TCLUST encounters difficulties when handling problems of increasing dimensionality, a situation that is becoming increasingly common in modern statistics. A possible solution is to consider methods that simultaneously address dimensionality reduction and clustering. In this context, and still within the trimming framework, RLG [3] is a useful alternative for higher-dimensional settings. Nevertheless, RLG may be somewhat simplistic and presents certain limitations that can be mitigated by combining it with TCLUST. In this talk, we present some numerical and graphical tools based on this combination of TCLUST and RLG, aimed at exploring atypical observations in settings with moderate to high dimensionality, beyond those where TCLUST and RLG are ideally suited. High-dimensional data introduce additional challenges. As dimensionality increases, there is also an increasing need for approaches capable of trimming not only unusual observations (row-wise trimming) but also atypical individual entries (cell-wise trimming). Methods have been introduced to integrate clustering with strategies to handle cellwise contamination (see [4] and [5]). Finally, for outlier detection under heterogeneity, it is sensible to initially discard a conservative fraction of the data, possibly larger than necessary. This high conservative initial trimming has the undesired effect of shrinking the dispersion estimators of the different components fitted by the clustering procedure. In fact, while estimating cluster centers and shapes is relatively straightforward, determining the clusters’ scales (which can be interpreted as “size” parameters of the components’ scatter matrices) is well known to present greater challenges [6]. However, accurate estimation of cluster scale parameters is essential for setting thresholds and measuring atypicality. It can be easily seen that scale estimation is closely interrelated with assessing the (unknown) level of contamination and the total weights of the cluster components. These quantities can be jointly estimated by following sequential iterative procedures [7], using a controlled sequence of trimming levels combined with robustified Mahalanobis distances. The radius process [8] provides a theoretical framework for these procedures and has been extended to accommodate distributions with heavier tails [9].

The Robust Interval Distributional Regression (RiDR) model represents a step forward in Symbolic Data Analysis (SDA) [1], evolving from the structural foundations of the ID model [2]. While traditional regression models often decompose intervals into separate linear regressions for bounds or midpoints and ranges, the ID framework treats intervals as holistic entities by representing them through quantile functions. By assuming a specific distribution within each interval – typically a Uniform distribution – the model accounts for internal variability that is lost in classical point-based summaries. The primary innovation of this framework lies in its mathematical constraints, which ensure that predicted outputs are always mathematically valid intervals. By incorporating both the quantile function of the independent variables and their symmetric counterparts, the model allows for both direct and inverse linear relationships through parameters a_(j) and b_(j), j = {1, …, p}. Crucially, the model imposes non-negativity constraints (a_(j), b_(j) > 0), which guarantees that the predicted range can never be negative, thus ensuring that the lower bound never exceeds the upper bound. Secondly, to address the sensitivity of standard least-squares methods to atypical data (outliers), the RiDR model integrates robust principles from the Exponential-type kernel robust regression (iETKRR) model [3]. It employs an objective function which uses Gaussian kernels [4, 5] to measure the similarity between observed and predicted values. Through an iterative re-weighting process, the model identifies and penalizes outliers by assigning them near-zero weights in an iterative way, effectively reducing their influence on parameter estimation in the midpoint and/or the range spaces. At each iteration, given these fixed weights, the model’s parameters are updated by solving a constrained quadratic programming problem that minimizes the objective function. This combination of distributional theory and robust kernel weighting allows the RiDR model to maintain high precision even in the presence of leverage points, X-space and Y-space outliers. Furthermore, the model remains highly flexible; it reduces to classical linear regression when applied to degenerate intervals (real numbers) and the weights settled as one, and can be adapted to alternative distributions, such as the Symmetric Triangular distribution, making it a versatile tool for complex aggregated datasets. Finally, the RiDR model and the non-robust counterpart were applied to real interval-valued data set [2] and the performances compared. The models were applied to the original data and the contaminated data set, where one observation from the original data set was replaced by an outlier. The results demonstrated that the RiDR model outperform the non-robust counterpart based on different metrics.

We consider numerical distributional data, where for each unit S_(i), i=1,…,n and each variable Y_(j) , j=1,…,p, a distribution is recorded, representing the corresponding observed variability. In our model, each distribution is represented by a set of indicators: a central statistic C, and the logarithm transformation of inter-quantile ranges, for a chosen set of quantiles ϕ₁,…,ϕ_(q), denoted by R_(h)^(*), h=1,…,m, where m=q−1 is the number of considered intervals. Typical cases consist in using the median, or else the midpoint, as central statistics, and quartiles, or other equally-spaced quantiles [1]; interval-valued data are represented by midpoints and log-ranges (m=1) [2]. Furthermore, we consider alternative structures of the variance-covariance matrix. In the most general formulation we allow for non-zero correlations among all central statistics and log-ranges; for distributional variables there are however other cases of interest, whether the variables, the central statistics, and the different log-ranges, are or are not correlated between or among themselves, leading to five different configurations in the distributional data case, and four configurations for interval-valued data. In this work we consider these data as matrix-valued data, represented as a tensor of dimension n×p×q, following [3], X ∼ ME(M,Σ_(var),Σ_(ind),g), where • Σ_(var) is p×p and gathers variances and covariances between the variables Y_(j) • Σ_(ind) is q×q and gathers variances and covariances between the considered q=m+1 indicators, C, R₁^(*),…, R_(m)^(*) • g(z) = exp(−z/2) / (2π)^(pq/2) In this model, the global covariance matrix Σ is written as Σ = Σ_(ind) ⊗ Σ_(var). This implies that we assume that covariances between the different indicators are constant across variables, thereby obtaining a more parsimonious model and reducing the number of parameters to be estimated. The different covariance configurations correspond to setting Σ_(ind) and/or Σ_(var) as block-diagonal matrices. The Matrix Minimum Covariance Determinant (MMCD) method [3] accounts for the matrix-variate data structure and robustly estimates the mean matrix, as well as the row-wise Σ_(var) and column-wise Σ_(ind) covariance matrices. Given n units S_(1, …, )S_(n ) ∈ R^(p × q), let Σ̂_(var) ∈ R^(p × p) and Σ̂_(ind) ∈ R^(q × q)  denote mode-wise estimators of the variables’ and indicators’ covariance matrices, respectively, obtained via either the matrix maximum likelihood estimator [4] or the matrix minimum covariance determinant method [3]. PCA is then performed independently on each estimator via eigen decomposition, Σ̂_(var)U = U Λ_(var) Σ̂_(ind)V = V Λ_(ind) (1) yielding variable-space loadings U ∈ R^(p×k) and indicator-space loadings V ∈ R^(q×ℓ). The matrix PCA scores are then Z_(i) = U^(T)(S_(i)− M̂) V ϵ ℝ^(k × l) i = 1,…,n (2) where M̂∈ R^(p × q) is an estimate of the mean matrix, providing a bilinear low-rank representation that captures variance along both matrix dimensions. This framework was applied to a sample of 1544 distributional data units of Spotify track data, collected from the Spotify’s public API, and aggregated by artist. The original microdata consisted of 170 663 track observations on 19 music attributes. For the purpose of this analysis we have considered nine numerical attributes (valence, acousticness, danceability, duration, energy, liveness, loudness, speechiness, and tempo) and nine distributional indicators (the median, and the log-ranges of the intervals defined by the min, max and the q_(0.125), q_(0.25), q_(0.375), q_(0.5), q_(0.75), and q_(0.875) quantiles) for each distributional data unit. The results lead to low-rank representations with interpretable and interesting components for both matrix dimensions. Furthermore, the comparison between the PCA results based on maximum likelihood and MMCD covariance estimates highlighted the impact of unit outliers, and the importance of robustly estimating covariance matrices in symbolic matrix-valued data.

Clustering Analysis is a fundamental tool in machine learning used to uncover underlying structures in unlabeled datasets by grouping similar instances together. While partitional methods like Fuzzy C-Means (FCM) are widely adopted for their ability to model overlapping classes through membership degrees, they face significant limitations in real-world scenarios [1]. Specifically, FCM is highly sensitive to noise and outliers because it relies on mean-based prototypes (centroids). Furthermore, standard fuzzy algorithms typically employ global distance metrics that treat all variables as equally important, which degrades performance in high-dimensional datasets containing irrelevant or correlated features. To address these challenges simultaneously, this work proposes the Multivariate Fuzzy C-Medoids (MFCMd) algorithm. The MFCMd method integrates the robustness of medoids [2] – actual observations from the dataset – with a multivariate membership mechanism [3] that performs local soft feature selection. Unlike centroids, a medoid is restricted to be an instance with the minimal aggregate dissimilarity to other members of the cluster, significantly improving interpretability and robustness against extreme values. The algorithm minimizes an objective function that accounts for distances between data points and cluster medoids across individual variables, enabling the estimation of feature relevance during the clustering process. Two distance-based variants are explored: MFCMd-E, utilizing the Euclidean distance, and MFCMd-C, utilizing the Cityblock distance. A significant contribution of this work is the extension of classical sum-of-squares-based interpretation indices [4] to the multivariate fuzzy context. These indices decompose total global dispersion into intra-group and inter-group components at the cluster and variable levels. This decomposition allows for a granular understanding of which variables contribute most to the distinction of specific groups. The computational complexity of MFCMd is O(c²p²n + cpn²) per iteration, reflecting the trade-off required to achieve simultaneous robustness to outliers and feature weighting. Experimental evaluations were conducted on seven synthetic configurations and seven real-world datasets, including Iris, Wine, and Cervical Cancer. Results demonstrated that MFCMd consistently outperforms traditional methods such as FCM, Fuzzy C-Medoids (FCMd), and the Gustafson-Kessel (GK) algorithm [5] in complex, non-spherical scenarios. Specifically, in high-dimensional or heterogeneous spaces, the ability to assign variable-specific weights allows the model to suppress noisy attributes and focus on features that define the cluster structures. In the Wine dataset analysis, the MFCMd-C variant successfully isolated prototypes with sharp chemical signatures that Euclidean-based methods failed to capture. Furthermore, using the proposed interpretation indices for variable selection led to significant performance improvements, with increases of up to 38.5% in the Adjusted Rand Index (ARI) [6] compared to using the full feature set. These findings confirm that the combination of medoid-based prototypes and multivariate learning provides a superior framework for uncovering complex, anisotropic patterns in modern data analysis.

Interval-valued data frequently arises in many real-world contexts where microdata are aggregated into macrodata, or where uncertainty and imprecision are inherent, leading to observations represented by intervals rather than single numerical values. Such data structure may exist by their own right or from aggregating original data due to security or convenience reasons. In this work, we consider the modeling framework introduced in Oliveira et al. [1], that links the macrodata with the underlying microdata distributions associated with each interval. Within this setting, we study the L1-Wasserstein distance and investigate its role as a comparison measure for interval-valued variables. This distance is defined originally in terms of the quantile functions of the microdata distributions associated to each interval. We present an explicit form of this distance, facilitating their use in data analysis. The performance of the L1-Wasserstein distance is evaluated through partitional clustering algorithms, including centroid-based and medoid-based approaches. The analysis also includes a comparison with Mallow’s distance (L2-Wasserstein), which has been previously studied as a basis for defining location and association measures [2], and as the underlying metric in Fisher discriminant analysis for interval-valued data [3]. The evaluation is based on simulation studies and further supported by real-data experiments. The empirical behavior and robustness of the clustering methods using this metric under contamination is examined through a structured simulation study in which three different scenarios are considered: outliers affecting the interval centers, the interval ranges, and a mixed configuration combining both components. The experimental design varies key structural factors including the number of clusters, the number of variables, the distributions, and the number of observations, among others. This analysis allows us to identify settings in which the L1 formulation exhibit improved performance, as well as scenarios where both approaches behave similarly. Overall, the study contributes to a clearer understanding of the behavior of the L1-Wasserstein metric when working with interval-valued variables. In particular, it characterizes its performance and robustness in clustering settings under contamination.

In the practical problems, we often measure a physical or economic entity Z, whose values can be influenced by its past values. Such situation is described with the autoregression model, with generally unknown autoregression parameters: Y_(t) =  θ₁Y_(t − 1)  + …  + θ_(p)Y_(t − p) + Z_(t), t = 1, …, n, where Y_(t − 1) , …, Y_(t − p) are the past measurements, while Z_(t) is the value of the interest. We repeat the experiment n-times with observations Y_(t1) , …, Y_(tn)and wish to make an inference on Z_(t1) , …, Z_(tn). For instance, we need to estimate the quantile function of Z. In another situation, Z_(t) can follow a linear regression model Z_(t)  = β₀ + Xβ + u_(t),  t = 1, …, n, while the autoregression is a nuisance. Then the problem is to estimate the parameter  β. Another time we compare two experiments, for instance the measurements of the amounts of CO₂, emitted from two different sources in the region. The measurements form two autoregressive sequences of orders p, q and leading to Z_(t), Z_(t)^(*). Then we want to verify the independence of two series {Z₁,…,Z_(n)} and {Z₁^(*). ,…,Z_(n)^(*). } by a rank test of independence, e.g. by the Spearman rank correlation coefficient. Our tools for such inference are the autoregression quantiles and the autoregression rank scores for the model. Because the AR rank scores are invariant with respect to changes of the autoregression parameters, the AR rank scores for Y_(t) coincide with those for Z_(t). We can replace the unknown ranks of Z_(t) with the regression rank scores, what considerably helps to solve the situation even with the unobservable Z_(t).

This talk will first discuss asymptotic relative efficiency (ARE) of a signed rank estimator in an errors in variables linear regression model with known Gaussian distributions of the measurement error, the predicting covariate and its surrogate. The ARE of this estimator relative to the bias corrected least squares estimator at a Gaussian regression error distribution is shown to increase to infinity as the measurement error variance increases to infinity. This is a kind of robustness property of this estimator against large measurement error variance. Given this motivation, we then derive asymptotic normality of an analog of this signed rank estimator in a class of nonlinear semi- parametric regression models where the predicting random covariate vector is possibly dependent on the regression error and where the regression error distribution need not be known.

We consider combined regression–autoregression models, motivated by time series with heavy tails, outliers and leverage effects. These models combine the influence of external covariates with autoregressive dependence on past observations, and therefore require inference methods that remain reliable under nonstandard innovation distributions. We propose a robust rank-based inference approach in which rank scores are calibrated using l-moment characteristics of the residuals. For a candidate parameter vector, residuals are computed and standardized by an l-scale estimate, providing stability under heavy-tailed innovations. L-moment ratios, namely l-skewness and l-kurtosis [1], computed from the standardized residuals are then used to adapt the score function, yielding a data-driven weighting scheme in Jaeckel’s dispersion criterion [2]. The resulting R–L estimator is obtained by minimizing this dispersion with adaptive scores, implemented through a simple reweighting iteration initialized by a Wilcoxon-score R-fit. Consistency and √n-asymptotic normality are discussed under standard regularity conditions, with the adaptive step treated as a plug-in procedure. Simulation results indicate that the proposed approach improves the mean squared error of regression coefficient estimates compared with ordinary least squares and classical fixed-score R-estimators, especially under contamination and heavy-tailed innovations.

M-quantile regression provides a flexible and robust alternative to classical regression by allowing the analysis of different parts of the conditional distribution of a response variable. Although this methodology has been widely studied under the independence assumption, serial dependence, such as autoregressive error terms, has not gained much attention in the literature. In many real applications, however, regression errors are correlated over time, and ignoring this structure may reduce efficiency and distort inference. In this study, we consider M-quantile estimation [1] in linear regression models with autoregressive errors. More specifically, the error term is assumed to follow an AR(p) process, enabling the model to capture serial correlation while preserving the robustness properties of M-quantile methods. The proposed framework extends standard M-quantile regression to situations where observations are ordered in time or where residual dependence cannot be neglected. An iterative estimation procedure is employed for the joint estimation of regression coefficients and autoregressive parameters. The method is expected to provide a robust modeling strategy in the presence of both outliers and autocorrelation. Its finite-sample performance can be examined through simulation studies under different sample sizes, dependence levels, and innovation distributions. Comparisons with conventional procedures may illustrate the practical advantages of incorporating autoregressive dependence into the M-quantile framework. Overall, the proposed approach offers a useful extension of robust regression methodology for dependent data settings.

In the classical one-way and two-way analysis of variance (ANOVA) procedures, distribution of the error terms is assumed to be normal and the observations are obtained via simple random sampling (SRS). However, in this study, logistic error distribution is considered, since its symmetric form and heavier tails make it a widely used and plausible alternative to the normal distribution. Also, ranked set sampling (RSS), originally proposed by McIntyre [1], is used as the sampling scheme due to its potential to yield more efficient estimators than SRS without increasing the sample size. Robust and efficient estimators of the unknown model parameters are obtained through modified maximum likelihood (MML) methodology, see Tiku [2], and then test statistics based on them are proposed. The performances of the proposed tests, shortly referred to as RSS based tests, are compared with those of the classical SRS based tests in terms of their sizes and powers through a comprehensive Monte Carlo simulation study. Particular emphasis is also placed on evaluating the robustness properties of RSS based tests in the presence of data anomalies and model misspecifications. In addition, the impact of imperfect ranking, which is frequently encountered in practical RSS applications, is investigated. Simulation results indicate that RSS based tests are more powerful than their SRS based counterparts besides having smaller type I error rates in most of the scenarios. Moreover, the proposed tests possess strong robustness properties to the deviations from the assumed model, and their performance advantage persists even under imperfect ranking conditions. A real data set is also analyzed to demonstrate the implementation of the proposed methodology.

In this study, ranked set sampling (RSS) based one-sample and two-sample tests are proposed as more efficient and robust alternatives to the corresponding tests existing in literature, see McIntyre [1] in the context of RSS. The error terms are assumed to follow logistic distribution, which is a reasonable alternative to the well-known normal distribution due to its symmetric shape. Logistic distribution has heavier tails than normal distribution and therefore is widely used for modelling data sets having outliers. The unknown model parameters are estimated by using the modified maximum likelihood (MML) methodology proposed by Tiku [2], which is based on the idea of linearization of the nonlinear terms in the likelihood equations and therefore yields closed-forms solutions. MML estimators are asymptotically equivalent to the maximum likelihood (ML) estimators besides being robust. Then, RSS based robust one-sample and two-sample tests based on these estimators are defined. Their performances are evaluated in terms of type I error and power via an extensive Monte Carlo simulation study. Simulation results demonstrate that proposed tests do not only have significantly higher power than their competitors in general but also maintain lower type I error rates. Robustness properties of the RSS based robust one-sample and two-sample tests are also studied to understand how they perform when there exist plausible departures from the assumed model. Based on numerical outcomes, the proposed tests are shown to exhibit greater robustness than their competitors against various violations of the model assumptions, including the presence of outliers, mixture and contamination models, as well as misspecification of the error distribution. Furthermore, the effect of imperfect ranking is investigated, and the findings reveal that the powers of the proposed tests are still remarkably high even in cases of severe errors in ranking when the set size is relatively small. Finally, a real data set is analyzed by using the proposed methodology for illustrative purposes.

The integration of Renewable Energy Sources makes electricity prices highly volatile, rendering deterministic point forecasts inadequate [1], [2]. In this context, uncertainty quantification of predictive models is crucial for decision-making problems [3], [4]. Conformal prediction (CP) offers a theoretically grounded framework for generating valid prediction intervals [5]. However, Inductive Conformal Prediction (ICP) fails in time-series because it relies on the assumption of exchangeability [6]. While several methods have emerged to adapt CP to non-exchangeable sequences, [7], [8], [9], Sequential Predictive Conformal Inference (SPCI) [10] has surfaced as a leading edge-case solution. Despite its strengths, the choice of the underlying “driver” model within the SPCI framework remains under-explored. To bridge this gap, we propose integrating a rolling Dynamic Multiple Quantile (DMQ) model [11]. This model not only calibrates residuals to react immediately to price shocks but also inherently prevents quantile crossing to ensure logical consistency. Furthermore, it maintains high performance even in the presence of market outliers, providing a robust foundation for uncertainty quantification. Evaluated on the German EPEX market, our proposed framework consistently outperforms standard benchmarks. Notably, it is the only model to uniquely pass the Kupiec test across all confidence levels, demonstrating superior calibration and reliability.

While adapting pre-trained foundation models via linear probing is a ubiquitous paradigm, the resulting models remain highly vulnerable to out-of-distribution (OOD) shifts [1] driven by spurious correlations [2, 3]. Traditional Domain Generalization (DG) methods attempt to solve this by learning invariant representations [4], but they frequently suffer from optimization instabilities, lack rigorous finite-sample guarantees, or underperform simple Empirical Risk Minimization (ERM) [5]. To address this fundamental challenge, we introduce Geometric Robustness Invariant Training (GRIT). By leveraging a small dataset of noisy invariant pairs—samples that share underlying semantics but differ in spurious attributes—GRIT estimates the spurious linear subspace of the feature embedding and forces the linear classifier to be orthogonal to this spurious subspace. Notably, this framework generalizes robustness in causal modeling: latent causal models with linear observations satisfy our spurious shift assumption—which generalizes classical covariate shift—while the concept of invariant pairs maps to spurious counterfactuals. Crucially, we provide a rigorous finite-sample statistical analysis of this framework. We prove that the out-of-distribution test risk explicitly decomposes into in-domain risk, a covariate shift term, and a spurious subspace misalignment term. Our theoretical guarantees demonstrate that the spurious term shrinks at a rate of $O(1/\sqrt{k})$ with k noisy invariant pairs, ensuring provable generalization even when the data pairs contain sub-Gaussian measurement noise. Empirical evaluations on CLIP embeddings [6] confirm that GRIT significantly improves worst-group accuracy over ERM and strong DG baselines on challenging benchmarks like Waterbirds [3] and ColoredMNIST [4]. [] Figure 1: GRIT Illustration – While ERM  $\theta\ \hat{}$ (with solid line decision boundary) fitted on the training domains (circles and triangles) is not robust to shifts in the spurious feature in the unseen test domain (pluses), the robust linear classifier $\theta\hat{}*$ (with dashed line decision boundary) can be obtained via our GRIT method by enforcing orthogonality to the difference vector of a single clean invariant pair (green line). Color indicates the label and shape indicates domain. Noisy pairs would require more than one pair but the error shrinks as $O(1/\sqrt{k})$ where k is the number of noisy invariant pairs.

As increasingly complex and powerful AI models have emerged, the need to understand and explain their behavior has grown, giving rise to the field of eXplainable AI (XAI). In the existing AI literature, Local Interpretable Model-agnostic Explanations (LIME), one of the most widely used post-hoc explanation methods alongside SHAP, provides local interpretations by fitting a surrogate model to randomly perturbed samples. Despite its pioneering role in the development of local explanation techniques, LIME suffers from limited reproducibility. This limitation arises from the way neighborhood samples are generated, which can produce unrealistic observations. Consequently, the correlation between the variables may not be adequately preserved. This can reduce the fidelity of the surrogate model in the neighborhood of the observation being explained. In this study, we propose a robust extension of LIME that incorporates copula-based sampling and gradient-informed weighting. These modifications aim to generate more realistic samples and yield more robust local explanations. According to the well-established benchmark metrics, simulation studies demonstrate that the proposed method remains highly consistent with the standard LIME framework while providing improved performance.

Functional Near-Infrared Spectroscopy (fNIRS) provides a non-invasive window into brain activity but is highly vulnerable to motion artifacts, sensor displacement, and physiological noise. These sources of contamination pose significant challenges for predictive modeling, particularly in real-world settings where signal quality is often compromised. Moreover, fNIRS data exhibit a complex hierarchical structure, with measurements collected across multiple channels nested within subjects and repeated over time. Subject-specific modeling and robust settings are therefore not merely a statistical refinement but a practical necessity for reliable inference in this domain. This study presents a comprehensive cross-paradigm robustness benchmark evaluating several models across five methodological families: pure statistical models for repeated measures (including Linear Mixed Model (LMM) and its robust counterpart), hybrid statistical-machine learning (ML) models, pure ML algorithms, deep learning architectures, and a temporal deep learning model. The benchmark targets prediction of standardized oxygenated hemoglobin change (ΔHbO) using an fNIRS dataset comprising 30 subjects and 20 channels across three experimental conditions [1]. LMM analysis established a significant hierarchical structure with nested random intercepts for Channel within Subject: the overall nested random-effects structure was highly significant. The nested channel-within-subject random intercept specifically contributed an additional significant improvement, while fixed effects explained negligible variance. This result motivates models capable of capturing individual-level temporal dynamics. Robustness was assessed via a Contaminate-Then-Split (CTS) protocol in which synthetic outliers of two magnitudes (±10 SD, ±15 SD) were injected at three contamination rates prior to temporal train-test splitting. Results reveal three principal findings. First, contamination magnitude governs predictive degradation more strongly than rate. Second, the Robust LMM demonstrates the most stable performance across all scenarios, validating M-estimation theory in practice [2]. Among pure ML models, Support Vector Regression proves most robust owing to the implicit outlier tolerance of its ε−insensitive loss function, whereas Extreme Gradient Boosting exhibits systematic overfitting in every scenario, achieving the lowest training error yet among the highest test errors across all conditions. Third, the proposed Dilated Temporal Network with Learned Random Effects (DilTNet-RE; see Figure 1), which combines dilated 1-D convolutions with learned Subject, Channel, Subject and Channel interaction embeddings, achieves the lowest test MAE on clean data, outperforming competing models by explicitly capturing slow hemodynamic autocorrelation across sequential trials that static models discard by treating observations independently [3]. DilTNet-RE maintains this ranking across all contaminated scenarios with a training-test error pattern consistent with good generalisation, though its margin over competing models remains modest. [] Figure 1: Architecture of the Dilated Temporal Network with Learned Random Effects (DilTNet-RE). Input sequences of 16 time steps with 6 features per step pass through positional encoding, a 6-block dilated convolutional stack (dilations 1–32), and global average pooling. The temporal summary is concatenated with learned Subject, Channel, and Subject×Channel embeddings before a two-layer prediction head produces the standardized HbO estimate.

The empirical likelihood method is a parameter estimation method that can be used with small sample sizes, possesses the flexibility of non-parametric methods, and exhibits the advantages of parametric likelihood methods [1]. Although this method inherently assigns a probability value to each observation, the constraints on the relevant parameter are based on classical methods—OLS for regression model parameters—therefore estimates are affected by data contamination and/or the presence of outliers. To cope with such situations, it is preferable to write the problem in the form of an exponential family or to use M-estimation-based constraints. While the exponential family formulation enhances the flexibility and numerical stability of the dual optimization process across various data structures, M-estimation-based constraints provide a direct defense mechanism by bounding the influence of severe outliers [2, 3]. In this study, it is aimed to compare different scenarios in terms of breakdown points using MSEs calculated to determine which method is more robust in estimating linear regression parameters under various sample sizes and contamination rates.

We propose a robust extension of finite mixture models of multivariate Wrapped Normal distributions[1] for the analysis of heterogeneous circular data on a p-dimensional torus. While finite mixtures of Wrapped Normal distributions provide a flexible framework for modelling multivariate circular or directional observations, standard maximum likelihood estimation may be severely affected by atypical observations, contamination, or model misspecification. To address this issue, we develop a weighted likelihood estimation procedure in which the contribution of each observation to the likelihood is adaptively weighted according to its agreement with the assumed model and with the mixture component membership. As for the non-robust case, parameter estimation is carried out through a nested weighted EM-type algorithm that exploits the unwrapping representation of circular data. As in the classical approach, the outer E-step accounts for the unobserved component memberships, whereas the inner E-step handles the latent vectors of wrapping coefficients and includes the additional weight for robustness. The proposed robust procedure preserves the flexibility of the original model while improving stability and resistance to outlying observations. The finite-sample performance of the method is investigated through Monte Carlo simulation studies under both clean and contaminated scenarios, and its practical usefulness is illustrated on real multivariate circular datasets.

This talk focuses on the methodological and computational challenges of optimizing Rényi’s $\alpha$-divergences within the framework of Variational Inference (VI). While standard VI relies almost exclusively on the Kullback-Leibler divergence, moving to the broader family of Rényi’s $\alpha$-divergences introduces optimization difficulties that require dedicated theoretical analyses as well as variance-reduction strategies [1,2].

In this talk, we will present a new statistical principle that combines the most desirable aspects of both parameter inference and density estimation. This leads us to the PrO posterior, which expresses uncertainty as a consequence of predictive ability. Doing so leads to inferences which predictively dominate both classical and generalised Bayes posterior predictive distributions. Our PrO posteriors adapt to the level of model misspecification: they concentrate around the true model in the same way as Bayes and Gibbs posteriors if the model can recover the data-generating distribution, but do not concentrate toward a single parameter in the presence of non-trivial forms of model misspecification. Instead, they stabilise towards a predictively optimal posterior whose degree of irreducible uncertainty admits an interpretation as the degree of model misspecification — a sharp contrast to how Bayesian uncertainty and its existing extensions behave.

In practice, planning a test to proof the product reliability while the development of a new product is a central task of reliability engineering. Based on a suitable quantitative reliability target, a common approach is to apply a Reliability Demonstration Test (RDT). The basic principle of RDT is based on successful testing when all test items survive the test duration without any failure [1]. Important test parameters can be determined based on the Binomial distribution for the so called success run [2]: 1 − CL= R_(min)(t_(target))^(n) (1) In practice, the test duration often does not correspond to the required product service life. In such cases, a service life ratio LR can be used, which describes the ratio between test duration t_(test) and service life requirement t_(target) [4]: $LR = \ \frac{t_{\text{test}}}{t_{\text{target}}}$ (2) Taking (2) into account in (1), the verifiable minimum reliability R_(min) under the assumption of a Weibull-distributed failure characteristic with a shape parameter β is thus as follows [1]: $R_{\min}\left( t_{\text{target}} \right) = \ \left( 1 – CL \right)^{\frac{1}{\text{LR}^{\beta}n}}$ (3) The shape parameter β is often estimated from failure data of similar products in operating conditions that are as comparable as possible. Another option within a product development project is the possibility to derive the shape parameter β from previous test phases of the product development project. The possible difference of the test specimens from different sample phases from design changes or manufacturing influences must be evaluated with regard to a changed failure characteristic [5]. These uncertainties in the estimation of the shape parameter β consequently lead to an erroneous determination of the necessary sample size or confidence level of an RDT [5], see Fig. 1. [] [] Figure 1: Effects of uncertain input data on sample size and confidence level of a RDT To improve the robustness of RDT, an enhanced approach of test planning by using Monte Carlo simulations is applied. Based on these simulations, both bias errors and uncertainties in the determination of the shape parameter can be quantified and taken into account on the basis of sample results [5]. [] Figure 2: Quantification of uncertainties of shape parameters based on Monte Carlo simulations

The global Multidimensional Poverty Index (MPI), jointly developed by UNDP and the Oxford Poverty and Human Development Initiative (OPHI), operationalizes poverty as a weighted sum of ten binary deprivation indicators across three domains: health, education, and living standards. Estimated from nationally representative household surveys across more than 100 countries, the global MPI is an important empirical tool for monitoring Sustainable Development Goals (SDG) 1.2, which commits to reducing poverty “in all its dimensions”. However, despite the MPI’s role in global poverty monitoring, forward-looking analysis remains underdeveloped; existing approaches rely on aggregate trend extrapolation¹ that is sensitive to sparse data and structural uncertainties, which are conditions endemic to low-income country contexts. This paper introduces a robust, scenario-based microsimulation framework that characterizes plausible future trajectories of multidimensional poverty to 2030, explicitly accounting for data limitations and structural uncertainty in underlying demographic processes. The analysis covered 71 countries that had relevant household survey data available for more than two time periods.
The framework is built around three interlocking statistical components. First, baseline household survey microdata are reweighted to align with projected 2030 population structures, preserving joint household characteristics under demographic uncertainty. Second, Latin Hypercube Sampling over logistic-bounded deprivation trajectories generates 1,200 scenario combinations per country, spanning plausible ranges of demographic and indicator-specific change. Third, intercept-shifted logistic regressions with Monte Carlo simulation translate these macro-level targets into household-level deprivation probabilities, yielding a full distribution of poverty outcomes. Representative scenarios are then extracted via machine learning classification models and diagnosed through ANOVA-based variance decomposition. In the first phase of the analysis, baseline household survey microdata is reweighted to reflect projected 2030 demographic structures, incorporating age composition, sex distribution, and rural-urban dynamics.² This demographic re-weighting preserves the joint distribution of household characteristics while accommodating uncertainty in population trajectories, addressing a significant gap in prior aggregate-level projection methods. In the second phase, we construct plausible indicator-specific trajectories for each of the ten MPI deprivation indicators at the country level. Annual rates of change are estimated via logistic transformation, which is appropriate for bounded proportions, and projected forward from the most recent observation to 2030. Recognizing that a single growth-rate estimate is unreliable given sparse time series and potential structural breaks,³ we introduce symmetric ±25% scaling of each estimated growth rate to define upper and lower bounds. A key robustness concern addressed here is the treatment of indicators with historically increasing deprivation rates: we evaluate three methodological options for handling these cases, including floor constraints, zero-growth caps, and a “pandemic tax” adjustment that widens uncertainty intervals for countries whose most recent data predate 2020. Sensitivity analyses confirm that cross-country variation in projected poverty is not driven by the length of inter-survey intervals, though outlier growth rates in 16 countries are identified and treated with additional country-specific rules. The third phase implements a household-level microsimulation linking macro-level indicator targets to individual household characteristics. For each indicator, weighted logistic regressions model baseline deprivation as a function of demographic covariates: household size, child composition, and rural location, and intercepts are adjusted so that predicted national prevalence matches each scenario’s indicator-specific target. Monte Carlo simulation is then applied, drawing 100 stochastic Bernoulli outcomes per household-indicator pair within each scenario, with results averaged to produce stable estimates. Crucially, the ten MPI indicators are treated as independent during scenario generation, avoiding the imposition of macroeconomic coherence assumptions that would be difficult to justify given the complexity and unpredictability of structural change. The 400 indicator scenarios, generated via Latin Hypercube Sampling to ensure stratified coverage of the joint input space, are combined with the three demographic variants to yield 1200 distinct, internally consistent projections per country. In the fourth phase, representative “prosperity” and “deprivation” scenarios are identified from the 1200 simulations using random forest classifiers for each country. Scenarios are classified relative to each country’s own distribution of simulated outcomes, ensuring that the representative trajectories reflect plausible upper and lower bounds for that country’s specific historical context rather than an absolute external standard. Applied to 71 countries with at least two survey rounds between 2003 and 2023, the framework produces several substantive findings relevant to the SDG period. Global multidimensional poverty remains heavily concentrated in Sub-Saharan Africa under both scenarios. The gap between prosperity and deprivation scenarios, ranging from one to eight percentage points across countries, provides a direct measure of each country’s sensitivity to alternative development pathways. This analysis makes key contributions to the robust statistics literature as applied to development measurement. First, it demonstrates how stratified sampling techniques (Latin Hypercube Sampling) and ensemble machine learning methods (random forests) can be combined with classical microsimulation⁴ to navigate high-dimensional uncertainty spaces efficiently and interpretably. Second, it formalizes a set of methodological decision rules for handling problematic time-series features: monotonically increasing trends, outlier growth rates, and pandemic-era discontinuities, within a transparent and replicable robustness framework. Together, these elements constitute a transferable methodological toolkit for robust scenario analysis in settings where data are scarce, structural breaks are plausible, and single-point predictions are epistemically unjustifiable. The framework is designed to be extensible: future iterations could incorporate macroeconomic coherence constraints, Bayesian updating of growth-rate priors as new survey data become available, or explicit conflict-adjustment modules for currently excluded fragile states. As the 2030 SDG deadline approaches and the limits of deterministic forecasting become increasingly apparent, methods that characterize the full distribution of plausible futures, rather than collapsing uncertainty into a single trajectory, offer the most defensible and policy-relevant basis for global poverty monitoring. Beyond the specific application to multidimensional poverty, the combination of Latin Hypercube Sampling, logistic-bounded uncertainty intervals, and random forest classification represents a broadly applicable template for scenario-based inference under data scarcity.

The precise identification of dental implants from radiographic imaging is a cornerstone of clinical assessment, restorative planning, and retrospective forensic records. While deep learning-based automated systems have proliferated, their clinical adoption is frequently hindered by the demand for massive, annotated datasets and their inherent “black-box” nature, which complicates clinical accountability. This study presents a comprehensive comparative analysis of three fundamental feature extraction techniques-Scale-Invariant Feature Transform (SIFT), Oriented FAST and Rotated BRIEF (ORB), and Accelerated-KAZE (AKAZE)-to establish a high-precision, interpretable framework for implant recognition in periapical radiographs. Utilizing a specialized dataset of 600 high-resolution periapical images, the study subjects these algorithms to a rigorous “stress test” framework, simulating clinical artifacts such as additive Gaussian noise, perspective rotations (±15⁰), and motion blur. Experimental results demonstrate that the SIFT-based approach achieves a deterministic success rate of 100%, outperforming AKAZE (96%) and ORB (91%). Statistical validation using McNemar’s test (p<0.001) confirms that SIFT’s superiority is mathematically significant and robust against acquisition variances. Unlike opaque neural networks, the proposed feature-based method offers “explainability maps,” allowing clinicians to visually verify correspondences in thread pitch and apex geometry. These findings suggest that SIFT provides the necessary robustness for a reliable clinical decision support system, offering a transparent and computationally efficient alternative for modern digital dentistry where data scarcity remains a significant challenge. The Scale-Invariant Feature Transform (SIFT), introduced by Lowe (2004), revolutionized this field by extracting keypoints that are highly resistant to changes in scale, rotation, and affine transformation. On the other hand, the ORB algorithm (Rublee et al., 2011) emerged as a computationally efficient, open-source alternative. By utilizing FAST keypoints and binary BRIEF descriptors, ORB offers significant advantages in terms of processing speed and memory usage, though it sometimes sacrifices the extreme robustness found in SIFT. In dental imaging, multi-scale matching is especially relevant. Implant features such as thread pitch, collar shape, and apex design exhibit consistent morphological signatures across scales. Studies in orthopedics and dental biomechanics (e.g., Wang et al., 2016) affirm the discriminative power of localized pattern comparison when supported by adequate normalization. To simulate real-world clinical conditions, images were subjected to a “Stress Test” comprising additive Gaussian noise, perspective rotations, and simulated motion blur. The performance was quantified using the Top-1 Identification Accuracy, defined as the exact matching of a sample to its corresponding database entry. Furthermore, the Distance Ratio (Lowe’s Ratio) was calculated to measure the confidence margin between the first and second best matches, providing a quantitative metric for “matching certainty”. The proposed SIFT-based identification system was executed on a standard workstation (Intel i5-12400, 16GB RAM). The average feature extraction and matching time per image was recorded at 142.5 milliseconds, making it suitable for real-time clinical integration without specialized hardware. The proposed methodology is designed to operate as a Clinical Decision Support System (CDSS) for dentists who encounter “orphan” implants with missing patient records. The system follows a four-stage execution pipeline: 1. Image Input: Direct upload of a periapical radiograph. 2. Feature Mapping: Real-time extraction of SIFT keypoints from the implant body. 3. Geometric Verification: Comparison against the local database of annotated implant models using a FLANN-based (Fast Library for Approximate Nearest Neighbors) matching engine. 4. Ranked Reporting: Output of the most likely implant models with a confidence score and visual “matching-lines” for clinician verification. The comparative results across all 600 images are summarized in Table 1. Our findings indicate a clear hierarchy in descriptor reliability for dental implant recognition. Table 1: Multi-Metric Comparative Performance Analysis Metric SIFT ORB AKAZE ——————————- ————— —————- ————– Top-1 Identification Accuracy 100%  91%  96% Avg. Keypoints per Image $842 \pm 124$ $3000$ (Fixed) $618 \pm 95$ Avg. Matching Time (ms) $142.5$ $18.2$ $64.7$ Lowe’s Ratio Margin ($\mu$) $0.42$ $0.78$ $0.58$ Precision 1.00 0.89 0.94 Recall 1.00 0.91 0.96 F1-Score 1.00 0.90 0.95 To evaluate the statistical significance of the performance disparity between SIFT and ORB, a McNemar’s test was performed. Given the 100% success rate of SIFT versus the 91% of ORB, the calculated p-value was found to be 0,001, indicating that SIFT’s superiority in periapical implant identification is statistically significant and not due to random variation in image quality or acquisition angle. This study has provided a rigorous evaluation of the clinical resilience and diagnostic precision of SIFT, AKAZE, and ORB descriptors for the automated identification of dental implants in periapical radiographs. Despite the exceptional results, it is acknowledged that the precision of the system is inherently linked to the quality of the initial periapical acquisition. While our stress tests confirm high robustness, extreme underexposure or anatomical overlap may still necessitate secondary imaging.

Circular (angular) data appear in many applications where observations are directional or periodic, such as wind directions, animal movement paths, and phase measurements. Since these data are defined on a circle, their analysis requires statistical methods that respect their periodic structure rather than standard linear approaches [1, 2]. A key problem in this setting is measuring the relationship between two angular variables. Several circular-circular correlation coefficients have been developed for this purpose [3, 4]. Although these methods are widely used, they are often sensitive to unusual observations. Even a small number of atypical data points can affect the results and reduce the reliability of the estimates. In this study, we focus on this issue and propose robust alternatives for circular-circular correlation. The proposed methods are based on Huber and Tukey weighting schemes, which limit the influence of extreme observations while keeping the main pattern in the data. The performance of the proposed methods is examined through an extensive simulation study under different sample sizes, contamination levels, and contamination structures. Within this framework, the new estimators are also compared with existing methods to better understand their behavior in both clean and contaminated data settings. Overall, this study aims to provide simple and reliable alternatives for correlation analysis in angular data, especially in situations where unusual observations are present.